A tethered jailbreak allows third-party apps and tweaks to function on
the device, although it is necessary to plug the device into a computer
running Redsn0w every time it is started up. In addition, newer iPhone,
iPod touch and iPad models aren’t compatible with this jailbreak, only
older Pre-A5 devices like the iPhone 4, iPhone 3GS and iPod touch
4th-generation are compatible.

1- Download iOS 6.1 and the original iOS 6.0 [Download iOS 6 from here], which is required for this jailbreak to work.
2- Download Redsn0w 0.9.15b3 for Mac or for Windows.

3- Head on over to Extras > Select IPSW and select the iOS 6.0 IPSW. Allow a few seconds until Redsn0w recognizes the file.

Note: The reason why we need iOS 6.0 for this to work on 6.1 is
because Redsn0w hasn’t been updated to officially work with iOS 6.1 yet.
4- Now, simply click on “Jailbreak” from Redsn0w’s main screen. On
the next screen, click “Next” and select “Cydia”. You then need to boot
your device in DFU mode and follow the on-screen instructions.
Redsn0w will now go on to do its work and jailbreak your device using
the Limera1n exploit. Cydia will then be installed. After the jailbreak
is completed, simply select the “Autoboot this device when it connects
in DFU mode” option. If you’ve gone this far, your device is now
jailbroken!How-to boot in tethered mode:
This is a tethered jailbreak, therefore your device needs to be
booted into a jailbroken state on every reboot. In order to do this, you
simply connect it to a computer running Redsn0w, navigate to “Extras”
and select “Just Boot”. Since most of us rarely shut down our devices
completely, you likely won’t need to do this often, unless there is a
power shortage.